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Octahedral Hole

The display below shows six atoms packed together in a closest-packed arrangement. This particular geometry is octahedral. In the very center of this octahedron is a hole. This hole is said to be a octahedral hole because the hole is surrounded by six atoms (the coordination number is six).

Take note that the size of this hole is smaller than that of the atoms surrounding it. If an atom is sufficiently small, it can fit into this hole. Click on the "Show Atom in Hole" to insert a small atom (colored red) in the hole. (You may need to rotate the image to clearly see the red atom in the tetrahedral hole.)

What is the largest atom that can fit into the hole without pushing the outer (blue) atoms apart?

The radius of the outer atoms is designated r. The radius of the hole (this is the radius of the largest sphere that can fit in the hole) is represented by rhole.

What is the ratio rhole/r ?

Derive this ratio using the octahedral geometry of the outer (blue) atoms and the laws of trigonometry.